1 /* poly/solve_cubic.c
2 *
3 * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2007, 2009 Brian Gough
4 *
5 * This program is free software; you can redistribute it and/or modify
6 * it under the terms of the GNU General Public License as published by
7 * the Free Software Foundation; either version 3 of the License, or (at
8 * your option) any later version.
9 *
10 * This program is distributed in the hope that it will be useful, but
11 * WITHOUT ANY WARRANTY; without even the implied warranty of
12 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 * General Public License for more details.
14 *
15 * You should have received a copy of the GNU General Public License
16 * along with this program; if not, write to the Free Software
17 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
18 */
19
20 /* solve_cubic.c - finds the real roots of x^3 + a x^2 + b x + c = 0 */
21
22 #include <config.h>
23 #include <math.h>
24 #include <gsl/gsl_math.h>
25 #include <gsl/gsl_poly.h>
26
27 #define SWAP(a,b) do { double tmp = b ; b = a ; a = tmp ; } while(0)
28
29 int
gsl_poly_solve_cubic(double a,double b,double c,double * x0,double * x1,double * x2)30 gsl_poly_solve_cubic (double a, double b, double c,
31 double *x0, double *x1, double *x2)
32 {
33 double q = (a * a - 3 * b);
34 double r = (2 * a * a * a - 9 * a * b + 27 * c);
35
36 double Q = q / 9;
37 double R = r / 54;
38
39 double Q3 = Q * Q * Q;
40 double R2 = R * R;
41
42 double CR2 = 729 * r * r;
43 double CQ3 = 2916 * q * q * q;
44
45 if (R == 0 && Q == 0)
46 {
47 *x0 = - a / 3 ;
48 *x1 = - a / 3 ;
49 *x2 = - a / 3 ;
50 return 3 ;
51 }
52 else if (CR2 == CQ3)
53 {
54 /* this test is actually R2 == Q3, written in a form suitable
55 for exact computation with integers */
56
57 /* Due to finite precision some double roots may be missed, and
58 considered to be a pair of complex roots z = x +/- epsilon i
59 close to the real axis. */
60
61 double sqrtQ = sqrt (Q);
62
63 if (R > 0)
64 {
65 *x0 = -2 * sqrtQ - a / 3;
66 *x1 = sqrtQ - a / 3;
67 *x2 = sqrtQ - a / 3;
68 }
69 else
70 {
71 *x0 = - sqrtQ - a / 3;
72 *x1 = - sqrtQ - a / 3;
73 *x2 = 2 * sqrtQ - a / 3;
74 }
75 return 3 ;
76 }
77 else if (R2 < Q3)
78 {
79 double sgnR = (R >= 0 ? 1 : -1);
80 double ratio = sgnR * sqrt (R2 / Q3);
81 double theta = acos (ratio);
82 double norm = -2 * sqrt (Q);
83 *x0 = norm * cos (theta / 3) - a / 3;
84 *x1 = norm * cos ((theta + 2.0 * M_PI) / 3) - a / 3;
85 *x2 = norm * cos ((theta - 2.0 * M_PI) / 3) - a / 3;
86
87 /* Sort *x0, *x1, *x2 into increasing order */
88
89 if (*x0 > *x1)
90 SWAP(*x0, *x1) ;
91
92 if (*x1 > *x2)
93 {
94 SWAP(*x1, *x2) ;
95
96 if (*x0 > *x1)
97 SWAP(*x0, *x1) ;
98 }
99
100 return 3;
101 }
102 else
103 {
104 double sgnR = (R >= 0 ? 1 : -1);
105 double A = -sgnR * pow (fabs (R) + sqrt (R2 - Q3), 1.0/3.0);
106 double B = Q / A ;
107 *x0 = A + B - a / 3;
108 return 1;
109 }
110 }
111